Flat solutions of the 1-Laplacian equation

For every $f \in L^N(\Omega)$ defined in an open bounded subset $\Omega$ of $\mathbb{R}^N$, we prove that a solution $u \in W_0^{1, 1}(\Omega)$ of the $1$-Laplacian equation ${-}\mathrm{div}{(\frac{\nabla u}{|\nabla u|})} = f$ in $\Omega$ satisfies $\nabla u = 0$ on a set of positive Lebesgue measure. The same property holds if $f \not\in L^N(\Omega)$ has small norm in the Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.Comment: Dedicated to Jean Mawhin. Revised and extended version of a note written by the authors in 201

Strong maximum principle for Schr\"odinger operators with singular potential

We prove that for every $p > 1$ and for every potential $V \in L^p$, any nonnegative function satisfying $-\Delta u + V u \ge 0$ in an open connected set of $\mathbb{R}^N$ is either identically zero or its level set $\{u = 0\}$ has zero $W^{2, p}$ capacity. This gives an affirmative answer to an open problem of B\'enilan and Brezis concerning a bridge between Serrin-Stampacchia's strong maximum principle for $p > \frac{N}{2}$ and Ancona's strong maximum principle for $p = 1$. The proof is based on the construction of suitable test functions depending on the level set $\{u = 0\}$ and on the existence of solutions of the Dirichlet problem for the Schr\"odinger operator with diffuse measure data.Comment: 21 page

Global regularity for systems with pp-structure depending on the symmetric gradient

In this paper we study on smooth bounded domains the global regularity (up to the boundary) for weak solutions to systems having $p$-structure depending only on the symmetric part of the gradient.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1607.0629

Suitable weak solutions to the 3D Navier-Stokes equations are constructed with the Voigt Approximation

In this paper we consider the Navier-Stokes equations supplemented with either the Dirichlet or vorticity-based Navier boundary conditions. We prove that weak solutions obtained as limits of solutions to the Navier-Stokes-Voigt model satisfy the local energy inequality. Moreover, in the periodic setting we prove that if the parameters are chosen in an appropriate way, then we can construct suitable weak solutions trough a Fourier-Galerkin finite-dimensional approximation in the space variables

Convergence Analysis for a Finite Element Approximation of a Steady Model for Electrorheological Fluids

In this paper we study the finite element approximation of systems of $p(\cdot)$-Stokes type, where $p(\cdot)$ is a (non constant) given function of the space variables. We derive --in some cases optimal-- error estimates for finite element approximation of the velocity and of the pressure, in a suitable functional setting

Convergence of approximate deconvolution models to the mean Magnetohydrodynamics Equations: Analysis of two models

We consider two Large Eddy Simulation (LES) models for the approximation of large scales of the equations of Magnetohydrodynamics (MHD in the sequel). We study two $\alpha$-models, which are obtained adapting to the MHD the approach by Stolz and Adams with van Cittert approximate deconvolution operators. First, we prove existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations. Then we study the behavior of solutions as the deconvolution parameter goes to infinity. The main result of this paper is the convergence to a solution of the filtered MHD equations. In the final section we study also the problem with filtering acting only on the velocity equation